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An Efficient Newton Algorithm for Nonnegative Matrix Factorization with the Kullback-Leibler Divergence

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Canonical URL: http://arxiv.org/abs/2607.13919v1
Topic: Research Misc
First seen: 2026-07-16 19:07:58
Last seen: 2026-07-16 19:07:58

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arXiv2026-07-16 19:06:50
An Efficient Newton Algorithm for Nonnegative Matrix Factorization with the Kullback-Leibler Divergence
Nonnegative Matrix Factorization (NMF) is a fundamental tool in unsupervised learning, which approximates a nonnegative matrix by the product of two low-rank nonnegative factors. The Kullback-Leibler (KL) divergence is best suited to measure the data to model discrepancy when the decomposed data sample follows a Poisson distribution, which is the case for count datasets such as term-document matrices or images. Most KL-NMF algorithms in the literature minimize a separable majorant of the loss to find their next iterate. We argue that this method has reached its limits and propose to use instead the second-order Taylor expansion of the loss, leading to a Newton-type method. We minimize this non-separable surrogate by proposing a generalization of the well-known HALS algorithm. This yields an efficient KL-NMF algorithm which provably converges and which competes favorably with state-of-the-art algorithms on a large variety of datasets.